Reasons for (prior) belief in Bayesian epistemology
Christian List(joint work with Franz Dietrich)
http://personal.lse.ac.uk/LIST/
Paper forthcoming in Synthese
LSE, November 2012
Introduction
Bayesian epistemology
tells us how to move from prior to posterior beliefs in light
of new evidence or information,
but says little about where our prior beliefs come from.
It offers few resources to describe some prior beliefs as
rational or well-justified, and others as irrational or
unreasonable.
Rational choice theory
A different strand of epistemology takes the central epistemological question
to be
not how to change one’s beliefs in light of new evidence (though this
obviously remains important),
but what reasons justify a given set of beliefs in the first place.
We offer an account of rational belief formation that closes some of the gap
between Bayesianism and its reason-based alternative.
We formalize the idea that an agent can have reasons for his or her (prior)
beliefs, as distinct from evidence/information in the Bayesian sense.
This is part of a larger programme of research on the role of reasons in
rational agency (Dietrich and List 2012a,b).
Posterior beliefs
Prior beliefs Evidence/information
FIXED/GIVEN CHANGEABLE
(Prior) beliefs
Credibility relation Doxastic reasons
FIXED/GIVEN CHANGEABLE
ALSO CHANGEABLEAs before
Posterior beliefs
Evidence/information
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
The objects of belief
We want to model how an agent forms his or her prior beliefs
over some set X of basic objects of beliefs.
The elements of X could be, e.g., possible worlds, states
of the world, or rival hypotheses.
We call them epistemic possibilities.
We only assume that the alternatives in X are mutually
exclusive and jointly exhaustive of the relevant space of
possibilities.
An agent’s beliefs
In Bayesian epistemology, the agent’s beliefs are usually
represented by a credence function (subjective probability
function) on X, which assigns to each possibility in X a real number
between 0 and 1, with a sum-total of 1.
However, we here begin by representing the agent’s beliefs by a
credence order on ≿ X (a complete and transitive binary relation).
x≿y means that the agent believes x at least as strongly as y.
( and ≻ denote the induced strict and indifference relations.)
Beliefs and belief formation
Bayesian epistemology gives an account of how an agent’s beliefs
should rationally change in response to evidence or information.
If the agent receives evidence that rules out some possibilities in X, he
or she must change the credence order so as to rank any possibilities
ruled out below (or weakly below) any possibilities not ruled out, while
not changing other rankings (Bayesian updating).
Here, however, we focus on the problem of belief formation:
How does the agent arrive at his or her beliefs over X in the first place,
before receiving any evidence?
Beliefs and belief formation
We can look at this problem from both positive and
normative perspectives, i.e., we can ask
either how an agent actually forms his or her beliefs,
or how he or she ought rationally to do so.
We develop a formal framework that can be used to
investigate both questions. This is where reasons come
into play.
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
Reasons in general
Reasons can be conceptualized in a number of ways.
Scanlon, e.g., defines a reason as “a consideration that counts in
favor of some judgment-sensitive attitude [e.g., belief or desire]”
(What we owe to each other, p. 67).
We adopt a more general definition, preserving the “counting” part
but not the “in favor” part of Scanlon’s definition.
We think of reasons as propositions that play a special role (that
somehow “count” or “matter”) in the agent’s attitude formation – in the
present context, in his/her belief formation.
Doxastic reasons
A proposition (in general) is a subset of X.
It is true of the possibilities contained in it, and false of all others.
More generally, propositions could be represented by sentences
from a suitable language.
We can also think of each proposition as capturing a particular
property of the epistemic possibilities.
Now suppose that there is some set of propositions, D, that the agent
focuses on in his/her belief formation process; we call these the
agent’s doxastic reasons.
Doxastic reasons
When a proposition is in D,
this does not mean that the agent believes it;
it only means that, in forming his/her belief about each epistemic
possibility, the agent considers whether or not the proposition
is true of that possibility.
So, the propositions in D stand for questions that the agent asks
him/herself in the process of belief formation.
We further define D to be the set of all possible such sets D.
D could simply be the set of all possible sets of propositions (or
smaller – I set technicalities aside).
The focal doxastic reasons
To indicate that the agent’s credence order ≿ depends on his
or her set D, we append the subscript D to the symbol ≿.
≿D = the agent’s credence order when D is focal
A full model of an agent’s beliefs requires the ascription of a
family (≿D)D D of credence orders to the agent, one ≿D for each
D D.
So, how exactly does the credence order ≿D depend on D?
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
An example: meeting in DC
Suppose I have agreed to meet Alexandru somewhere in Washington DC at
12 noon tomorrow.
We have not agreed on a place, and we have no way to communicate.
I have no evidence in the standard sense as to where Alexandru is likely to
expect me.
Here are some possibilities:
Union Station
Lincoln Memorial
White House
Hilary Clinton’s apartment
How do I form my prior beliefs over where Alexandru might expect me?
An example: meeting in DC
The following are some possible considerations that might be relevant:
A : The place in question is where one arrives in Washington ({Union}).
F : The place in question is world-famous ({Lincoln, WH}).
R : The place in question has restricted access ({WH, Hilary C.’s apt.}).
My credence orderings across variations in D might look like this:
D={A,F,R} Union ≻D Lincoln ≻D WH ≻D Hilary Clinton’s apt.
D={A,F} Union ≻D Lincoln D WH ≻D Hilary Clinton’s apt.
D={A,R} Union ≻D Lincoln ≻D WH D Hilary Clinton’s apt.
D={F,R} Lincoln ≻D WH ≻D Union ≻D Hilary Clinton’s apt.
D={A} Union ≻D Lincoln D WH D Hilary Clinton’s apt.
D={F} WH D Lincoln ≻D Union D Hilary Clinton’s apt.
D={R} Union D Lincoln ≻D WH D Hilary Clinton’s apt.
Union D Lincoln D WH D Hilary Clinton’s apt.
An example: meeting in DC
Can we say something systematic about these
beliefs?
They are what we call reason-based.
Reason-based beliefs
The agent’s family of credence orders is reason-based if there exists a
binary relation over sets of reasons (a credibility relation) such that:
for any D D and any x, y X:
The agent believes x more than y when focusing on the reasons in D
if and only if
the set of reasons in D that are true of x is ranked above
the set of reasons in D that are true of y;
formally, x ≿D y {RD : xR} {RD : yD}.
The example again… The following are a few possible considerations that might be relevant:
A : The place in question is where one arrives in Washington ({Union}).
F : The place in question is world-famous ({Lincoln, WH}).
R : The place in question has restricted access ({WH, Hilary C.’s apt.}).
My credence orderings across variations in D might look like this:
D={A,F,R} Union ≻D Lincoln ≻D WH ≻D Hilary Clinton’s apt.
D={A,F} Union ≻D Lincoln D WH ≻D Hilary Clinton’s apt.
D={A,R} Union ≻D Lincoln ≻D WH D Hilary Clinton’s apt.
D={F,R} Lincoln ≻D WH ≻D Union ≻D Hilary Clinton’s apt.
D={A} Union ≻D Lincoln D WH D Hilary Clinton’s apt.
D={F} WH D Lincoln ≻D Union D Hilary Clinton’s apt.
D={R} Union D Lincoln ≻D WH D Hilary Clinton’s apt.
Union D Lincoln D WH D Hilary Clinton’s apt.
These beliefs are reason-based, w.r.t. the following credibility relation:
{A} > {F} > {F,R} > > {R}
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
Two axioms on the relationship between reasons and beliefs
Axiom 1. ‘Principle of insufficient reason.’
For any x, y X and any D D,
if {R D : x R} = {R D : y R},
then x D y.
Axiom 2. ‘Invariance of relative likelihoods under the addition of
irrelevant reasons.’
For any x, y X and any D,D’ D with D D’,
if for all R D’ \D, x R and y R,
then x ≿D y x ≿D’ y.
The basic representation theorem
(1) The agent’s family of credence orders satisfies Axioms 1
and 2
if and only if
(2) it is reason-based.
That is, there exists a credibility relation over sets of
reasons such that, for any D D and x, y X,
x ≿D y {RD : xR} {RD : yR}.
Overview of this talk
Beliefs
Reasons for belief
An example
An axiomatic characterization result
The cardinal case
Beliefs: the cardinal case
We now go beyond considering credence orders and represent
an agent’s beliefs in the more standard way by credence
functions (subjective probability functions).
A credence function is a function Pr : X [0,1] such that
xX Pr(x) = 1.
(Probabilities of propositions are defined in the usual way.)
We write PrD as the agent’s credence function when D is his or
her set of doxastic reasons in relation to the possibilities in X.
We are interested in how PrD depends on D.
Two axioms on the relationship between reasons and beliefs
(the cardinal case)
Axiom 1. ‘Principle of insufficient reason’
For any x, y X and any D D,
if {RD : R is true of x} = {RD : R is true of y},
then PrD(x) = PrD(y).
Axiom 2. ‘Invariance of likelihood ratios under the addition of
irrelevant reasons’
For any x, y X and any D, D’ D with D D’,
if no R in D’ \D is true of x or y,
then PrD(x)/PrD(y) = PrD’ (x)/PrD’ (y).
Theorem 2
(1) The agent’s family of credence functions PrD across
all D D satisfies Axioms 1 and 2
if and only if
(1) there is a credibility function from possible
combinations of doxastic reasons into the real
numbers such that, for all D D and all x X,
({RD : R is true of x}) PrD(x) = .
x’ X ({RD : R is true of x’ })
Remarks
The (prior) probability of a given possibility is proportional to the
credibility of the set of doxastic reasons that are true of that
possibility.
The factor of proportionality,
1 / x’ X ({RD : R is true of x’ }) ,
depends on D and ensures that probabilities add up to 1.
Interpretationally,
just as practical reasons are good-making features of actions,
so doxastic reasons are plausible-making features of epistemic
possibilities.
(Prior) beliefs
Credibility relation Doxastic reasons
FIXED/GIVEN CHANGEABLE
ALSO CHANGEABLEAs before
Posterior beliefs
Evidence/information